In our two previous notes on this topic, we analyzed stress testing methodologies for default probabilities in “Bank of America and CCAR 2016 Stress Testing: A Simple Model Validation Example” and “An Introduction to Stress Testing: Oil Prices, Default Probabilities and Credit Spreads.”

We explained that there are five potential methodologies for linking macro-economic factors to default probabilities for the purpose of stress testing according to the Federal Reserve’s Comprehensive Capital Analysis and Review 2016 scenarios, scenarios of the European Banking Authority, or scenarios for other internal or third party purposes. The five potential methodologies are listed here:

- Use the same logistic regression formula and inputs used to determine the current term structure of defaults and historical defaults
- Use the lagged default probability as an input for a simple naïve model to predict forward default probabilities
- Use current macro factors to predict current default probabilities
- Use current macro factors and financial statement ratios and stock price inputs known at time zero to predict future default probabilities.
- Use current macro factors, forecasted financial statement ratios and forecasted stock price inputs to predict future default probabilities.

- An ordinary least squares regression fitted to the transformed default probabilities
- A fractional regression using the logistic distribution fitted to the actual unannualized default probabilities, expressed as a decimal.

**Champion and Challenger Models for Predicting Future Default Probabilities**

In this section we discuss all five methods estimation strategies and the two econometric procedures for executing those strategies. We start with Method 1 and Method 5, because both of these approaches are immediately rejected on both theoretical and practical grounds. Method 1 uses the same logistic formula and inputs for current and historical default probability generation (i.e. in sample) for out of sample default probability forecasting. The Kamakura Risk Information Services Version 6.0 Technical Guide lists the explanatory variables as inputs for the Jarrow-Chava reduced form model. Twenty-seven of those inputs are related to company-specific financial statements, stock prices or both. Since 27 out of 47 inputs are unknown on future dates, for this reason alone Method 1 is rejected.

**A Discussion of Methods 2, 3, and 4 using KRIS 3 Month Default Probabilities on Wells Fargo & Co. quarterly through September 30, 2015.**

- Financial statement inputs and stock price information from time zero, not from the future points in time.
- The firm’s default probabilities at time zero, not from the future points in time.
- Macro factors which will be known at the future points in time.

- Maximum likelihood estimation
- Fractional regression, logistic (a special case of maximum likelihood estimation)
- Non-linear least squares
- Ordinary least squares on the transformed default probability

**Background on the Ordinary Least Squares Approach**

The naïve regression explains 54.93% of the variation of Wells Fargo’s transformed 3 month default probabilities on a quarterly basis from January 1990 through September 2015. The error term is implicitly assumed to have a mean of zero and a standard error (deviation) of 0.88595 (note that this is the standard error for the transformed Z[t] variable, not the raw default probability itself). The lagged transformed default probability is statistically significant. When we reverse the transformation of Z[t], the equation we use to simulate the 3 month default probability one quarter forward is this:

**Method 4: Adding Time Zero Company Specific Explanatory Variables**

**Comparing Method 3 and Method 4**

**IMPORTANT CONCLUSION: Fractional regression (logistic) succeeds in an effective challenge of a more conventional approach, using ordinary least squares on the transformed default probabilities. Fractional regression is clearly the best practice approach from an accuracy point of view.**

**Implied CCAR Stress Test Results for Wells Fargo & Co.**

**Conclusion**

**References**

Copyright ©2016 Donald van Deventer